You'd need to handle setting the image path in update() just like you are in the create method. You're not just dumping $request to create. You're manipulating the data before you send it to create. You need to do the same in edit, or at least for the image_path variable.
// make an array of all request data except for the image path
$data = $request->except('image_path');
// now set the image path just like in create.
$data['image_path'] = request()->file('image')->store('images', 'public');
// update
$camping->update($data);
I'd actually redo your create() method too. Notice how much simpler it is to do
$data = $request->except('image_path');
$data['image_path'] = request()->file('image')->store('images', 'public');
$camping = Camping::create($data);
vs your
$camping = Camping::create([
'title' => request('title'),
'en_title' => request('en_title'),
'state' => request('states'),
'address' => request('address'),
'body' => request('body'),
'en_body' => request('en_body'),
'phone' => request('phone'),
'email' => request('email'),
'website' => request('website'),
'opening' => request('opening'),
'image_path' => request()->file('image')->store('images', 'public')
]);
There is more info about what you can do with the request object in the user guide if you care to learn more.